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Questions tagged [stochastic-processes]

For questions about stochastic processes, for example random walks and Brownian motion.

A stochastic process is a collection of random variables representing the evolution of a system of random variables over time. A typical example is a .

A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Sufficient conditions to specify a stochastic process?

In order to specify a stochastic process, is it sufficient to specify all the finite-dimensional distributions of the stochastic process? Can there exist two stochastic processes that agree in all the marginal distributions, but are not equal?
user2808118
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Will a process with mean step 0 cross every bound infinitely often?

Take a discrete time process with mean step size 0. The steps will take values > ε with positive probability. It seems intuitive that, for any bound, the process will cross the bound infinitely many times. Does anyone know of a result that says…
firepanda
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Expected generation of extinction in branching process with binomial offspring

Consider a branching process with immediate offspring distribution $\xi \sim \operatorname{Bin}(m, p)$, where $m$ is a constant. Let $\phi(s)$ be the generating function of $\xi$, i.e. $\phi(s) = (1 - p + s p)^m$, and let $\phi_n(s) :=…
d125q
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Poisson Processes question

Let $\{N(t) : t \geq 0\}$ be a Poisson process with rate $\lambda\gt 0$. Let $Y$ be a random variable independent of $N(t)$, such that $Y = 1$ with probability $1/2$ and $Y = −1$ with probability $1/2$. We define the new process $X(t)$ by $X(t) =…
user220502
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How to find the z-transform of a given quantity?

The arrival of messages to a communications channel is modeled as a Poisson process, with rate $\lambda$ messages / unit time. Let $\{N(t), t \geq 0\}$ denote that process. Each message contains a random number of bytes; the probability mass…
Mathematics Lover
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Prove $m(t)$ exists in renewal process

I have one question about the example of renewal process. For a renewal process, $N(t)+1$ is a stopping time for the interarrival sequence $X_1,X_2,\cdots$. Show that $\mathbb{E}[N(t)] < \infty$, i.e, check that $m(t) < \infty$. When I search for…
NiubilityDiu
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Calculate the pdf of $Z[n] = 3/4^{(n-1)}X[1] + 3/4^{(n-2)}X[2] + ... + 3/4X[n-1] + X[n]$.

Calculate the pdf of the sum $Z[n] = 3/4^{(n-1)}X[1] + 3/4^{(n-2)}X[2] + ... + 3/4X[n-1] + X[n]$. Where $X[n]$ is a $IID$ gaussian stochastic process with $mean=0$ and $variance =1$. Thanks!
Junior
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Hitting times of closed vs. open sets

If you have a continuous stochastic process, then how is the hitting time for a closed set different from the hitting time of an open set when we're trying to show it is a stopping time?
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Markov Chains: Expected Return Time (Stochastic Process)

I am given a matrix with space {0,1,2,3,4}. I already calculated the invariable probability vector. However, the question asks to give the expected number of steps: -given Xo=0 to go back to state "0". -given Xo=0 to go back to state "3". For Xo=0…
sahimat
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Need a little bit of guidance with stochastic processes

Let $X(t) = \begin{bmatrix} cos(t) + N(t)\\ sin(t) + S(t)\\ \end{bmatrix} $ (where $N(t)$ is a gaussian process and S(t) is a Poisson's process). Let $Y(t)=\begin{bmatrix} 2 & 4\\ 6 & 8\\ …
Lugi
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Continuou Time Markov Chains - Poisson Distribution

Suppose $X_t$ and $Y_t$ are independent Poisson processes with parameters $\lambda_1$ and $\lambda_2$, respectively, measuring the number of calls arriving at two different phones. Let $Z_t=X_t+Y_t$. (a) Show that $Z_t$ is a Poisson process. What…
Ninja
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Asymptotic behaviour of absolute different of two independent Poisson processes

Suppose we have $X_1,X_2\sim Po(\lambda)$ ($X_1$ and $X_2$ are independent). Consider the interval $[0,1]$ with 100000 subintervals of length $\Delta=\frac{1}{100000}$. I can…
Ocean
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Self similar process

I am learning long memory process and came cross the definition of self similar. By definition, process $X(t)$ is self similar if $X(at)=_d a^H X(t)$,$a>0$ and $H$ is Hurst exponent. By equality of all finite dimensional distributios, does this…
Ocean
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Doob–Meyer decomposition

Let $X_n = \sum_{m≤n} 1_{B_m} $and suppose $B_n \in F_n$. What is the Doob decomposition for $X_n$ ? I can write it down from the construction of the theorem, but is there any neat way showing the result?
annimal
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Covariance of a Bernoulli process

I have a Bernoulli process $\Phi(t)$ with a symmetric distribution $p=1/2$. The random variable can take values $a,b$. My question is what is the covariance of this process $\langle\Phi(t)\Phi(t')\rangle$? Thanks.
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